Why is expected bond energy the *geometric* mean of two pure bonds?

Question

Suppose we are calculating the electronegativity of a bond A—B. In my textbook (Chemical Principles) calculation of electronegativity, they calculate electronegativity in terms of a variable difference in bond energies $\Delta$, which they define as $$\ce{\Delta = (A-B)_{actual} - (A-B)_{expected} }$$

and that discrepancy is used to calculate electronegativity. Fair. But I'm confused by their formula for the expected energy of $\ce{A-B}$: $$\ce{Expected A-B bond energy = \sqrt{\ce{(A-A bond energy) \times (B-B bond energy)}} }$$

which is the geometric mean of the two. But why the geometric mean, and not, say, an arithmetic or harmonic mean? Why even use a mean at all? What's the intuitive rationale for this?

Answer 1

You can actually used arithmetic mean instead of geometric mean. Mean (or average) of 2 different value are designed to be somewhere in between the value.

For example, let $X=3$ and $Y=7$, the means are $$HM = \frac{1}{2}(\frac{1}{3} + \frac{1}{7}) = 4.2$$ $$GM = (3\times 7)^{\frac{1}{2}} \approx 4.58$$ $$AM = \frac{3+7}{2} =5$$ There are other types of mean that can also be use (i.e., root mean square).

Now, I believe that the electronegativity that you are referring to are Pauling's electronegativity. I think the main reason that Pauling pick $GM$ over the other is due to the fact that energy may come in many order of magnitude. Let's say $X=1$ and $Y=10000$ $$HM = \frac{1}{2}(\frac{1}{1} + \frac{1}{10000}) = 20000/10001 \approx 1.9998$$ $$GM = (1\times 10000)^{\frac{1}{2}} =100$$ $$AM = \frac{1+10000}{2} =5000.5$$

If you see this, which of the 3 means ($HM$, $GM$, $AM$) is more in the middle. It would be more intuitive to consider $GM$ to be more in the middle in this case.